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Mirrors > Home > MPE Home > Th. List > nnexpcl | Structured version Visualization version GIF version |
Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
Ref | Expression |
---|---|
nnexpcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 12248 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | nnmulcl 12267 | . 2 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 · 𝑦) ∈ ℕ) | |
3 | 1nn 12254 | . 2 ⊢ 1 ∈ ℕ | |
4 | 1, 2, 3 | expcllem 14070 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 (class class class)co 7420 ℕcn 12243 ℕ0cn0 12503 ↑cexp 14059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-exp 14060 |
This theorem is referenced by: digit1 14232 nnexpcld 14240 faclbnd4lem3 14287 faclbnd5 14290 climcndslem1 15828 climcndslem2 15829 climcnds 15830 harmonic 15838 geo2sum 15852 geo2lim 15854 ege2le3 16067 eftlub 16086 ef01bndlem 16161 phiprmpw 16745 pcdvdsb 16838 pcmptcl 16860 pcfac 16868 pockthi 16876 prmreclem3 16887 prmreclem5 16889 prmreclem6 16890 modxai 17037 1259lem5 17104 2503lem3 17108 4001lem4 17113 ovollb2lem 25430 ovoliunlem1 25444 ovoliunlem3 25446 dyadf 25533 dyadovol 25535 dyadss 25536 dyaddisjlem 25537 dyadmaxlem 25539 opnmbllem 25543 mbfi1fseqlem1 25658 mbfi1fseqlem3 25660 mbfi1fseqlem4 25661 mbfi1fseqlem5 25662 mbfi1fseqlem6 25663 aalioulem1 26280 aaliou2b 26289 aaliou3lem9 26298 log2cnv 26889 log2tlbnd 26890 log2ublem1 26891 log2ublem2 26892 log2ub 26894 zetacvg 26960 vmappw 27061 sgmnncl 27092 dvdsppwf1o 27131 0sgmppw 27144 1sgm2ppw 27146 vmasum 27162 mersenne 27173 perfect1 27174 perfectlem1 27175 perfectlem2 27176 perfect 27177 pcbcctr 27222 bclbnd 27226 bposlem2 27231 bposlem6 27235 bposlem8 27237 chebbnd1lem1 27415 rplogsumlem2 27431 ostth2lem3 27581 ostth3 27584 oddpwdc 33974 tgoldbachgt 34295 faclim2 35342 opnmbllem0 37129 heiborlem3 37286 heiborlem5 37288 heiborlem6 37289 heiborlem7 37290 heiborlem8 37291 heibor 37294 expgcd 41894 dvdsexpnn0 41901 hoicvrrex 45944 ovnsubaddlem2 45959 ovolval5lem1 46040 fmtnoprmfac2lem1 46906 fmtno4prm 46915 perfectALTVlem1 47061 perfectALTVlem2 47062 perfectALTV 47063 bgoldbachlt 47153 tgblthelfgott 47155 tgoldbachlt 47156 blenpw2 47651 nnpw2pb 47660 nnolog2flm1 47663 |
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